Optimizing Balance and Sample Size in MatchingMethods for Causal Inference∗

Gary King† Christopher Lucas‡ Richard Nielsen§

July 16, 2013

Abstract

We propose a greatly simplified approach to matching for causal inference that si-multaneously optimizes both balance (between the treated and control groups) andmatched sample size. This procedure resolves two widespread (bias-variance tradeoff-related) tensions in the use of this powerful and popular methodology. First, cur-rent practice is to run a matching method that maximizes one balance metric (suchas a propensity score or average Mahalanobis distance), but then to check whetherit succeeds with respect to a different balance metric for which it was not designed(such as differences in means or L1). Second, current matching methods either fix thesample size and maximize balance (e.g., Mahalanobis or propensity score matching),fix balance and maximize the sample size (such as coarsened exact matching), or arearbitrary compromises between the two (such as calipers with ad hoc thresholds ap-plied to other methods). These tensions lead researchers to either try to optimizemanually, by iteratively tweaking their matching method and rechecking balance,or settle for suboptimal solutions. We address these tensions by first defining thefrontier in the balance-sample size trade-off as the set of matching solutions withmaximum balance for each possible sample size. Researchers can then choose one,several, or all matching solutions from the frontier for analysis. Computation is fastand requires no iteration or manual tweaking. We offer easy-to-use software that im-plements these ideas and present a range of empirical analyses that demonstrate theirvalue.

∗Our thanks to Carter Coberley, Stefano Iacus, Giuseppe Porro, and Aaron Wells.†Albert J. Weatherhead III University Professor, Institute for Quantitative Social Science, 1737 Cam-

bridge Street, Harvard University, Cambridge MA 02138; http://GKing.harvard.edu, king@harvard.edu,(617) 500-7570.

‡Ph.D. Candidate, Institute for Quantitative Social Science, 1737 Cambridge Street, Harvard University,Cambridge MA 02138.

§Assistant Professor, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge MA02139; http://www.mit.edu/∼rnielsen, rnielsen@mit.edu, (857) 998-8039.

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1 IntroductionMatching has attained popularity among applied researchers as a powerful, intuitive, and

conceptually simple method of improving causal inference in observational data analysis.

It is especially simple as a nonparametric preprocessing step that identifies data subsets

from which causal inferences can be drawn with greatly reduced levels of model depen-

dence (Ho et al., 2007). Although successful applications of matching require both re-

duced imbalance (between the treated and control groups) and a sufficiently large matched

data subset, existing matching methods optimize with respect to only one of these two fac-

tors, with the required joint optimization performed by manually tweaking existing meth-

ods or ignored altogether. This is crucial since, if the subset identified by the matching

method is too small, the reduction in model dependence (and hence bias) achieved will be

counterbalanced by an unacceptably high variance. Similarly, the small variance associ-

ated with a matching method identifying a large data subset may be counterbalanced by

unacceptably high levels of imbalance (and thus model dependence and bias). Some of

this problem is also induced by current matching methods which optimize with respect to

one balance metric, whereas researchers using them verify levels of balance achieved or

adjusted post hoc with respect to a different metric for which the method was not designed

and cannot optimize.

To remedy these problems, but to still keep its advantage in simplicity, we introduce a

procedure that enables researchers to define, estimate, visualize, and then choose from the

matching frontier which characterizes the trade-off between imbalance (using any chosen

metric) and matched sample size. Like all matching methods, our approach attempts to

identify latent experiments in observational data. Unlike other methods, we allow re-

searchers to simultaneously evaluate how “experimental” (balanced) a matched sample

is and trade this off against the need for a matched sample size that provides sufficiently

precise estimates. At each location along the matching frontier, analysts can be confident

that they are getting the subset of the complete data that is most balanced relative to all

possible subsets at a given subset size. Similarly, as analysts move along the frontier, they

can be confident that the matched data subsets are becoming increasingly balanced as fast

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as possible. Any matching solution not on this frontier is suboptimal in that a lower level

of imbalance can be achieved for the same size data subset. Thus, our method achieves all

of the benefits of any individual matching method, allows researchers to extract maximal

causal information from their observational data, and avoids many of the pitfalls common

in current matching applications.

We begin by defining our notation, quantities of interest, the advantages of reducing

imbalance, imbalance metrics, and existing matching methods (Section 2). We then dis-

cuss problems with choosing a matching solution (Section 3) and then define our concept

of the matching frontier, along with algorithms for how to estimate it (Section 4). Last,

we compare our approach to existing practices in the context of the analysis of several

real data sets (Section 5).

2 DefinitionsWe define here our notation, causal quantities of interest, estimation, commonly used

matching methods, and frequently used imbalance metrics.

Notation For unit i (i = 1, . . . , n), let Ti denote a treatment variable coded 1 for units

in the treated group and 0 in the control group. Let Yi(t) (for t = 0, 1) be the (potential)

value the outcome variable would take if Ti = t. By definition, for each i, either Yi(1) or

Yi(0) is observed, but never both (which is known as the “fundamental problem of causal

inference”; Holland 1986). This means we observe Yi = TiYi(1) + (1 − Ti)Yi(0). Even

though only one potential outcome is ever observed, both must exist — which is implied

by the assumptions of common support, 0 < Pr(T = 1|X) < 1 (Heckman, Ichimura and

Todd, 1998, p.263), and stable unit treatment value, which can be thought of as logical

consistency, so that each potential value is fixed even if T changes, no interference, and

no versions of treatments (VanderWeele and Hernan, 2012).

Finally, we collect a vector of k pre-treatment control variables Xi to meet the condi-

tions of “strong unconfoundedness.” This condition requires that the treatment be inde-

pendent of the potential outcomes: T⊥{Y (0), Y (1)}|X (Rosenbaum and Rubin, 1983),

which requires (SU-X) having the right X so that this expression is possible and (SU-

control) controlling sufficiently for X so that it actually does hold. Virtually all methods

3

of observational data analysis, including all methods of matching and all modeling ap-

proaches, require SU-X as is; they then distinguish themselves by how they implement

approximations to SU-control.

Quantities of Interest Denote the treatment effect of T on Y for unit i as TEi =

Yi(1) − Yi(0). Most studies limit the estimation to averages of TE over relevant sub-

sets of units. One such quantity of interest is the sample average treatment effect on the

treated, SATT = 1nT

∑i∈{T=1} TEi, which is the treatment effect averaged over all (nT )

treated units.

If matching only prunes data from the control group, SATT is fixed throughout the

analysis. However, since most real data sets contain some observations without good

matches, often because common support between the treated and control groups do not

fully coincide, best practice has been to compute a causal effect among only those treated

observations for which good matches exist. We designate this as the feasible sample

average treatment effect on the treated or FSATT.1 Following most of the literature, our

analyses use FSATT, and so let the quantity of interest change, but everything described

below can be restricted to a fixed quantity of interest SATT, chosen ex ante, by always

retaining all treated units and only pruning control units.

Imbalance Measures v Metrics An imbalance measure is the degree of difference be-

tween the multivariate empirical densities of X for the treated and control groups. It

has been defined in practice in many different ways. For our purposes, we distinguish

imbalance measures from an imbalance imbalance metric which we define as a func-

tion d(X0, X1), given a matched data set with control X0 and treated X1 covariate vec-

tors, that satisfies four criteria: (1) nonnegativeness, d(X0, X1) >= 0; (2) symmetry,

1This decision is somewhat unusual in that the quantity of interest itself is defined by the statisticalprocedure, but in fact it follows the usual practice in observational data analysis of collecting data andmaking inferences only where it is possible to learn something. The advantage here is that the methodologymakes a contribution to ascertaining where this search will be fruitful (e.g., Crump et al., 2009; Iacus, Kingand Porro, 2011); one must only be careful to characterize the units being used to define the new estimand.As Rubin (2010, p.1993) puts it, “In many cases, this search for balance will reveal that there are membersof each treatment arm who are so unlike any member of the other treatment arm that they cannot serve aspoints of comparison for the two treatments. This is often the rule rather than the exception, and then suchunits must be discarded. . . . Discarding such units is the correct choice: A general answer whose estimatedprecision is high, but whose validity rests on unwarranted and unstated assumptions, is worse than a lessprecise but plausible answer to a more restricted question.”

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d(X0, X1) = d(X1, X0), or in other words replacing T with 1−T will have no effect; (3)

triangle inequality, d(X0, X1)+d(X1, Z) ≥ d(X0, Z), for some other k-vector Z; and (4)

monotonicity, given X0 and X1 with some level of imbalance, removing the worst match-

ing stratum of treated and control observations (leaving the matched set of treated and

control covariate vectors X ′0 and X ′

1 respectively) cannot increase the overall imbalance,

d(X0, X1) ≥ d(X ′0, X

′1). Requirements (1)–(3) define a generic (semi)distance metric;

we added (4) to ensure that the function is coherent when pruning observations from the

data set, which is how matching works. Imbalance measures which are metrics are usu-

ally decomposable so that the overall measure is a function of the sum over strata of the

results of applying the measure to each stratum of treated and control units.

We also go one step further and define “perfect balance” in two ways. In both, we

require strong unconfoundedness to hold, without additional assumptions about the data

generation process. In particular, we assume, as all other observational data methods,

that SU-X holds, and then we achieve SU-control in one of two ways, leading to the

two definitions. In the first, perfect balance requires that the distribution of treated and

control units in a data set has identical empirical support, meaning that, for each point

representing a treated unit in the k-dimensional data cloud, there exists at least one control

unit. The TE for the stratum of units at each of these points can be estimated using a

simple difference in means between treated and control units, ignoring X . Aggregating

up to SATT requires weighting, with the stratum TE weighted according to the number

of treated units in the stratum. Equivalently, the entire data set can be weighted such that

all treated units receive weights of W = 1, and all control units receiving weights of

W = (mT=0/mT=1) ∗ ((msT=1)/(msT=0

)) where mT=0 and mT=1 are respectively the

number of control and treated units in the data set, and msT=1and msT=0

are the number

of treated and control units in each stratum.2

Weighting is a simple procedure involving no additional assumptions, but it slightly

complicates subsequent data analysis. Thus, for some purposes we may wish to restrict

the concept of “perfect balance” to require that for each treated unit in k-dimensional

space there be j control units, such that k and j are fixed over the units i. In this situation,

2See j.mp/CEMweights for further explanation on weights.

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the multivariate empirical densities of treated and control units are identical, without need

for weighting in the subsequent analyses.

We usually prefer the first definition of perfect balance as identical empirical distri-

butional support of treated and control units because the further restriction of requiring

identical numbers of treated and control units at each stratum generally requires unnec-

essarily throwing away some observations that would lend precision without increasing

imbalance. However, an important goal of matching is simplicity and keeping users, and

so retaining the ability to tell them that they can preprocess with matching and do not need

to modify their existing procedures by weighting may be useful for some. It also turns out

to be true that the matching frontier algorithm we develop below for continuous distance

metrics naturally optimizes the first definition of perfect balance, while our algorithm for

discrete metrics naturally optimizes the second, although of course either definition can

be used with any imbalance metric.

Examples of Imbalance Measures We consider five types of matching measures. The

first two are decomposable metrics, which is easy to see because of the summation over

strata. The others are not.

First are continuous metrics such as the Mahalanobis discrepancy. For any two obser-

vations Xi and Xj the Mahalanobis distance is M(Xi, Xj) =√

(Xi −Xj)′S−1(Xi −Xj),

where S is the sample covariance matrix of X . Then the discrepancy is the Mahalanobis

distance between each unit i and the closest unit in opposite group, averaged over all units:

M = 1N

∑ni=1 M(Xi, Xj(i)), where Xj(i) = minj∈{1−Ti}M(Xi, Xj), where {1−Ti} is the

set of units in the treatment group that does not contain i, and N is the size of the observed

or matched data set being evaluated.

A second type of imbalance metric directly measures the difference between the mul-

tivariate histograms of the treated and control groups, defined by a chosen fixed bin size

H . Let f`1···`kbe the relative empirical frequency of treated units in bin with coordinates

`1 · · · `k, and similarly for g`1···`kamong control units. Then let

L1(H) =1

2

∑(`1···`k)∈H

|f`1···`k− g`1···`k

| (1)

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and

L2(H) =

√1

2

∑(`1···`k)∈H

(f`1···`k− g`1···`k

)2 (2)

To remove the dependence on H , define L1 as the median value of L1(H) from all possible

bin sizes H in the original unmatched data (Iacus, King and Porro, 2011); we use the same

value of H to define L2. The typically numerous empty cells of each of the multivariate

histograms do not affect L1 and L2, and so the summation in (1) and (2) each have at most

only n nonzero terms.

Third is based on the common approach of using the difference in means of each vari-

able j: d(j) = X̄(j)−X̃(j), where X̄(j) and X̃(j) are the means of variable j in treated and

control groups, respectively. The individual differences in variable means (perhaps after

rescaling) are then averaged either via absolute values, D1 = 1J

∑Jj=1 |d(j)|, or root mean

squares, D2 =√

1J

∑Jj=1 d(j)2. The fact that this measure is not decomposable to the

unit level, leads to the paradoxical result that at some point imbalance can increase even

after pruning the worst possible matching stratum (violating monotonicity). Violations of

monotonicity with these measures commonly occur in applications.

A fourth type of imbalance measure compares the distributions of propensity scores

across treatment conditions after matching. As with the Mahalanobis discrepancy, we

can calculate the propensity score discrepancy as the average difference in the estimated

propensity scores between each unit i and the closest unit in opposite group, averaged over

all units. This metric is conditional on the propensity score model choice being correct

and, as such, those who use propensity score matching almost always use some other

metric to evaluate the procedure’s success.

A final type of imbalance measure does not satisfy monotonicity and so may be useful

for a fixed sample size but not work for our purposes. One is the measure used in genetic

matching which “minimize[s] the largest individual discrepancy, based on p-values from

KS-tests and paired t-tests for all variables that are being matched on” (Diamond and

Sekhon, 2012, p.9). The other is the related d2 measure (Hansen and Bowers, 2008).

Nonmonotonicity is violated in part because they are based in part on p-values, which

change as a function of both sample size and imbalance (see Imai, King and Stuart, 2008).

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The Advantages of Reducing Imbalance In a perfectly balanced data set, the multi-

variate empirical distribution of X in the treated and control units are the same, and so

M = L1 = L2 = 0. In this situation, we do not need to condition on X; the two-part

strong confoundedness assumption simplifies SU-X; and a simple difference in means

is an unbiased estimator, obviating the need for difficult-to-justify statistical modeling

assumptions involving the k-dimensional X . Or in more simple language, since X is un-

related to T , it is not a confounder in the matched subset and we need not to worry about

how to control for it further.

Matching methods selectively prune observations, seeking to reduce the degree of

imbalance. If some but not all imbalance is eliminated, matching requires some statisti-

cal modeling assumptions but is typically very effective at reducing the dependence on

high dimensional modeling assumptions (Ho et al., 2007). The procedure is to take each

observation from the treated group (i.e., where Yi(1) = Yi is observed) and impute the

unobserved potential outcome Yi(0) with an observed unit (or units) j from the control

group such that it is matched to observation i either via “exact matching,” so that Xi = Xj ,

or, as is usually necessary in finite samples, “approximate matching”, so that Xi ≈ Xj

(with the definition of “approximate” varying across matching methods and defined be-

low). Unmatched observations are pruned from the data set before further analyses. Exact

matching produces balance; approximate matching reduces imbalance.

Finally, after matching, a statistical estimator is applied to the matched data set. For

the specific mathematical relationship between imbalance, model dependence, and bias,

see King and Zeng (2006) and Imai, King and Stuart (2008).

Matching Methods The most common implementation of Mahalanobis distance match-

ing (MDM) and propensity score matching (PSM) fix the number of observations that will

result from matching to twice the number of treated units; they work by matching each

treated unit to the nearest control unit, using that method’s chosen distance metric (usu-

ally via one-to-one nearest neighbor greedy matching without replacement; Austin 2009,

p.173). MDM measures the distance between the two observations via the Mahalanobis

distance, M(Xi, Xj). In PSM, we first collapse the vectors to a scalar “propensity score,”

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which is the estimated probability that an observation receives treatment given the covari-

ates, usually an estimate of a logistic regression, πi ≡ Pr(Ti = 1|X) = 1/(1 + eXiβ);

then, the distance between observations with vectors Xi and Xj is the simple scalar differ-

ence between the two estimates π̂i− π̂j (or sometimes Xiβ̂−Xjβ̂). The overall imbalance

achieved by this procedure is determined after the fact by examining the resulting matched

data. A related but more general procedure, genetic matching, also fixes the matched

sample size ex ante but through a computationally intensive algorithm enables the choice

among several imbalance metrics, the value of which is determined as an output of the

procedure (Diamond and Sekhon, 2012).

Coarsened Exact Matching (CEM) works by choosing a level of imbalance ex ante

and seeing how many observations are left at the end. Imbalance is fixed by temporarily

coarsening each covariate. (For example, years of education might be coarsened for some

purposes into grade school, high school, college, and post-graduate.) Then, units with the

same values for all the coarsened variables are placed in a single stratum. And finally, for

further analyses, control units within each stratum are weighted to equal the number of

treated units in that stratum, with observations in strata without at least one treated and

one control unit dropped. (Each treated unit weight is set to 1; control unit weights are

set to the number of treated units in its stratum divided by the number of control units in

the same stratum, normalized so that the sum of the weights across strata equals the total

matched sample size.) The unpruned units with the original uncoarsened values of their

variables are passed on to the analysis stage. See Iacus, King and Porro (2011, 2012).

Sometimes procedures are applied to the MDM and PSM matched data sets to remove

treated units unreasonably distant from (or outside the common support of) the control

units to which they were matched. The most common such procedure is calipers, which

are chosen cutoffs for the maximum distance allowed (Stuart and Rubin, 2007; Rosen-

baum and Rubin, 1985). This procedure then starts with whatever level of imbalance was

produced by the matched data set, reduces it by a specific known amount, but then the

number of observations remaining is only known after the fact.

Many refinements of these and other matching procedures have been proposed, only

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a few of which are widely used in applied work (for reviews, see Ho et al., 2007; Stuart,

2010).

3 Choosing a Matching SolutionCriteria To choose or evaluate the results a matching method, researchers confront the

same bias-variance trade-off as in most of statistics. However, two issues prevent one

from optimizing on this scale directly. First, because matching is a preprocessing step,

rather than a statistical estimator, particular points on the bias-variance frontier cannot be

computed without also simultaneously evaluting the estimation procedure applied to the

resulting matched data set. Second, best practice in matching involves avoiding selection

bias by ignoring the outcome variable while matching (Rubin, 2008), the consequence

of which is that we give up the ability to control either bias or variance directly. Thus,

instead of bias, scholars focus on reducing the closely related quantity, imbalance (for the

specific mathematical relationship between the two, see Imai, King and Stuart, 2008), and

instead of variance we focus on the small matched sample size. Optimizing with respect

to only imbalance or matched sample size, but not both, would be a mistake.

Existing Approaches in Practice Various largely manual (and self-described ad hoc)

procedures have been suggested in the methodological literature to try to compensate for

current matching methods which do not simultaneously optimize with respect to both

imbalance and matched sample size (e.g., Austin, 2008; Caliendo and Kopeinig, 2008;

Rosenbaum, Ross and Silber, 2007; Stuart, 2008). For example, Rosenbaum and Rubin

(1984) detail their “gradual refinement” of an initial model by including and excluding

covariates until they obtain a final model with 45 covariates, “including 7 interaction

degrees of freedom and 1 quadratic term”. Ho et al. (2007, p.216) write “one should

try as many matching solutions as possible and choose the one with the best balance.”

Imbens and Rubin (2009) propose a procedure to run PSM, check imbalance, adjust the

specification, and automatically rerun until convergence. Applying most of these methods

can be inconvenient and difficult to replicate. With MDM and PSM in particular, tweaking

the procedure to improve imbalance with respect to one variable will often make it worse

on others, and so the iterative process can be frustrating to apply. Presumably as a result,

10

applications of these (best practices) in applied literatures is uncommon.

4 Constructing a Matching FrontierWe now show how to simultaneously optimize with respect to both imbalance and the

number of pruned observations. We do this by constructing a matching frontier, which

is the set of matching solutions for which no other solution has lower imbalance for a

given sample size or larger sample size for given imbalance. Matching solutions not on

the frontier should not be used in applications (at least not without further justification)

since they are strictly dominated in terms of imbalance and number of observations by

those on the frontier.

Strategy In our approach, the user chooses an imbalance metric, which then imme-

diately defines the frontier (which the algorithms we offer below can generate). An

imbalance measure that is not an imbalance metric could be used, but then the frontier

will sometimes have counterintuitive properties, such as when removing observations in-

creases imbalance. Computation would also be greatly complicated. We therefore recom-

mend confining the choice to an imbalance metric.

Among the solutions along the frontier we construct, one can reasonably choose one

based on the matching version of the bias-variance trade-off: For example, in a large data

set, we can afford to prune a large number of observations in return for lower imbalance

(since our confidence intervals will still be fairly narrow), but in smaller data sets or when

an application requires especially narrow confidence intervals we might prefer a solution

on the frontier with fewer pruned observations at the cost of having to build a model to

cope with imbalance left after matching. If we choose to let matching help define FSATT,

rather than fixing the quantity of interest as SATT, the choice of a solution on the frontier

also helps us choose the estimand. Our algorithms will work with either choice.

We represent the frontier graphically, in which the number of pruned observations is

plotted on horizontally and a measure of imbalance is plotted vertically. We plot each

frontier solution, creating a series of points showing the trade-off between balance and

sample size. Since in this approach the two key optimization criteria are directly included

from the beginning, the common annoyance of checking imbalance or the matched sample

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size after matching is no longer necessary. All the results, and all relevant non-suboptimal

matching solutions that might be chosen, appear on the frontier in this one graph.

Algorithms The algorithms we propose to construct the matching frontier begin with

an imbalance metric that can be applied to each stratum of treated and control units and

then aggregated to produce the overall value. The algorithm works by simply removing

the stratum with the highest level of imbalance, and then the next highest, etc. The two

algorithms we discuss here work with and discrete imbalance metrics. In Table 1, we

present the algorithm for continuous distance metrics using Mahalanobis distance as an

example, but the algorithm also works for any other continuous distance metric, such as

based on Euclidean distances. Using the distance metric provided, our algorithm identifies

the nearest neighbor(s) in the opposite treatment condition and calculates the distance

from each unit to its nearest neighbor. We then remove units in order of distance, from

the longest distance to the shortest, until either the remaining distances are zero or there

are no units left. When two or more units are equally distant, as in the case of treated and

control units that are mutually closest to each other, we remove them simultaneously. This

algorithm optimizes toward a balanced data set in which the different treatment conditions

have common empirical support. Multiple identical matches are retained, so analysts will

need to use weights to calculate quantities of interest such as the FSATT.

Our second algorithm, shown in Table 2, produces a frontier according to the L1 and

L2 metrics. These metrics rely on discrete distances between potential matches — units

with X values that fall within the same bin of the multivariate histogram H specified by

the analyst are considered sufficiently proximate to match while all other are not. Un-

like the algorithm for continuous metrics, the L1/L2 frontier algorithm optimizes toward

a data subset in which the treated and control groups are identical in empirical distribu-

tion (up to the binning created by H), rather than simply having identical support. This

algorithm can be used to produce a data set that has identical empirical support, but non-

identical distributions by omitting step 3 from the algorithm and reweighting as described

in Section 2.

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Table 1: Mahalanobis Matching Frontier algorithm.1. Define the Aggregate Mahalanobis Discrepancy (AMD) between thetreated (T ) and control (C) groups:

a. Compute the minimum Mahalanobis distance (MMD) (using the kvariables in Xi) for each observation as its distance to the nearest unitin the opposite treatment regime (T for C or C for T )

b. Compute AMD as an average over these n distances.

2. Repeat until exit:

a. a← max of MMD vector.

b. Remove all observations (either T or C) with MMDi > a.

c. Compute and record AMD, the number of observations remaining(m), and (if desired) a quantity of interest.

d. If AMD = 0 or m = 0, stop

3. Draw frontier as a plot of AMD by m

Table 2: L1/L2 Matching Frontier algorithm.1. Define the L1 imbalance metric (as per Iacus, King and Porro 2011)

a. define bins based on the median of all possible binnings based on thecross-product of the coarsened covariates — in the raw data

b. Compute 12

of the sum of the absolute value of the differences acrossbins in the treated (T ) and control (C) groups.

2. Remove any bins with only T or C units. 3. Repeat until exit:

a. Record the value of L1 and the number of observations (m) remaining

b. If L1 = 0 or m = 0, stop

c. Prune one observation from the bin for which the T and C groupsdiffer the most (if the L1 criterion is indifferent between removingtwo observations, we use the one that optimizes with respect to L2)

4. Draw frontier as a plot of L1 by m

Uses The result of running either of these algorithms on a data set is a set of matched

samples that optimally trades off balance and sample size, given a specified imbalance

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metric. Analysts can use these matched samples in several ways. In some cases, it may

be desirable to select a single matched data subset on the frontier and use this subset to

calculate the FSATT. The distributions of each covariate in the matched subset can be

compared to the distributions in the original data to see what types of units were retained

and thus what the new quantity of interest is. Analyzing a single subset may be desirable

when analysts wish to produce a single estimate and have strong reasons to pick a partic-

ular point on the frontier (perhaps because imbalance goes to zero while the sample size

remains large). Of course, the analyst may also fix the quantity of interest to SATT, by

slightly adjusting the algorithm to only drop control units.

An alternative is to estimate the FSATT for every matched subset on the frontier and

to display them all. In conjuction with a characterization of the changes the value of the

covariates in the matched sample along the fontier — perhaps using the means of each

covariate, or a series of parallel plots — this procedure provides every reasonable causal

estimate that can be obtained using matching. Analysts may want to discount estimates

coming from regions of the frontier where imbalance remains high or where the sample

size decrease to the point that precise estimates are not longer possible. However, calcu-

lating the FSATT at each point on the frontier allows analysts to see how their quantity

of interest changes if they are willing to accept more or less imbalance in the matched

subset. It may be useful to select a single result from among the many estimated FSATTs

for more detailed discussion of substantive effects; readers with the full frontier available

to them can also easily assess whether the specific result highlighted is representative of

the other results along the frontier.

Our approach is similar to other work in matching that attempts to overcome many of

the problems with traditional matching applications discussed above. We build directly

on King et al. (2011), which identified the importance of the matching frontier, but was

unable to develop an algorithm to produce optimal frontiers; Diamond and Sekhon (2012)

which left the matched sample size fixed but offered the first automated method of reduc-

ing imbalance; Hansen (2004) which used full matching with restrictions; and many other

ad hoc approaches.

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5 ApplicationsWe now apply our approach to three real datasets. The first is a published get-out-the-

vote (GOTV) field experiment conducted in Iowa and Michigan before the 2002 midterm

elections, previously used to gauge the effectiveness of matching techniques (Arceneaux,

Gerber and Green, 2006). This study is especially useful, as it provides a valid instrument

through which the experimental result can be recovered and compared to the estimates

generated by our frontier approach. In the second study, we use the well-known Lalonde

dataset (LaLonde, 1986) in a similar comparison between an experimental result and one

generated via matching. Finally, we calculate the frontier for a published paper employ-

ing propensity score matching (Nielsen et al., 2011), and compare the results across the

methods.

For illustrative purposes, we report the L1 frontier and the Mahalanobis Discrepancy

frontier for the GOTV experiment, the L1 frontier for the Lalonde dataset, and the Maha-

lanobis frontier for Nielsen et al. (2011). For the latter two illustrations, the results for the

L1 frontier and the Mahalanobis Discrepancy frontier do not substantively differ.

5.1 Large-Scale Voter Mobilization Experiment

In (Arceneaux, Gerber and Green, 2006), households were randomly assigned to either

the treatment or control condition, and the treated units received a get out the vote (GOTV)

phone call encouragement to vote. Subjects in the experiment were recorded as having

received the treatment if they listened to the caller read a script and responded to the

question, “Can I count on you to vote next Tuesday?” (The response to the question

was irrelevant to the experiment.) In the experiment, units were split into four strata,

within which the treatment was randomly assigned. Those strata are competitive and

uncompetitive districts in Iowa and Michigan. For computational simplicity, we analyze a

random sample of 4,000 from the full dataset of 2,474,927, which was drawn conditional

on strata. The representation of each stratum is proportional to that in the full dataset, as

is the proportion of treated to control units within each stratum.

These data are especially interesting because they can be treated as either observa-

tional or, in the sense that no modeling assumptions are necessary, experimental. The

15

observational perspective comes from the fact that not all respondents assigned to the

treatment group listened to the full script or answered the phone, and so the households

that answered the phone and listened to the full script may differ from those which did not.

We can therefore estimate the treatment effect with either instrumentation or matching.

While a scenario with such a clean (randomly assigned) instrument obviously suggests

that one ought to prefer an instrumental variables estimator, it also creates a valuable op-

portunity to benchmark a matching estimator against an experimental result. To be clear,

however, this comparison is not an evaluation of our frontier method so much as it is a

test of both the extent to which the method controls well for X and the SU-X assumption

of strong ignorability; failing the test does not implicate the approach or the assumption

separately. Nevertheless, we offer this comparison here as part of the continuing effort of

methodologists to learn from observational data. In this light, we offer the estimate from

the experimental benchmark, using the instrumental variable estimator, which is −0.025

(with a standard error of 0.057), meaning that the estimate of the causal effect of a phone

contact is -2.5 percentage points in voter turnout.

L1 Frontier To begin, we provide the L1 frontier in Figure 1. The monotonicity and

general smoothness of the frontier is a theoretical property of the L1 frontier.

After calculating the frontier, we have the matched set with the lowest imbalance for

each possible subset size of the original sample, based on a notion of balance premised

on L1/L2. In this particular example, we keep the quantity of interest fixed and so do not

drop treated units. Therefore, the best possible balance with this data set is 0.062 (i.e.,

where the curve in Figure 1 ends at the right), where the remaining imbalance is due to

treated units in strata without common support. It is often desirable to drop treated units

as well as control, but for the purposes of replicating the instrumental variables estimator,

for this example, we leave all treated units in the sample and drop only controls.

Given that we now have the best possible matched data set for each number of obser-

vations, we can estimate the treatment effect for each point along the entire frontier. We

do this in Figure 2. After examining the frontier of estimates, it is clear that the differ-

ence between the experimental estimate of −0.025 (marked with a horizontal blue line)

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0.25

0.50

0.75

0 1000 2000 3000Number of Control Units Pruned

L1 V

alue

L1 Imbalance Frontier with GOTV Data

Figure 1: The L1 frontier with GOTV data, beginning with n = 4, 000.

and the initial OLS estimate of 0.035 (the point furthest left in Figure 2) is due to those

controls in strata without treated units (the vertical red dashed line marks the point in the

frontier at which all such units have been dropped). Thus, a researcher would be wise to

choose some dataset along the flat portion of the frontier to the right of the vertical red

dashed line. For a very long stretch in this range, at least until almost all the observations

have been pruned, the matching estimator is within range of the (instrumental variables)

experimental benchmark.

Mahalanobis Frontier Although in practice, a researcher would normally only use one

imbalance metric and thus have one frontier, for illustrative purposes we also compute

the frontier based on average Mahalanobis discrepancy imbalance metric, using the same

sample; see Figure 3.3 (The fact that the shape of this frontier differs from that for the L1

frontier in Figure 1 is solely due to the mathematical differences in the definition of the

3A property of this algorithm is that each iteration may remove more than one unit, when different unitshave identical Mahalanobis distances. (This contrasts with the L1 algorithm, in which units are removed oneat a time across all iterations.) To mark this difference, we draw the Mahalanobis frontier with individualpoints and connect them for visual clarity. The points represent values at which there is a defined datasetalong the frontier.

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●

●

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●●●

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●

●

●

●

●

●

●

●

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●

●

●

−0.05

0.00

0.05

0.10

0 1000 2000 3000Number of Control Units Pruned

Effe

ct S

ize

and

SE

s

Treatment Effects with GOTV Data Along L1 Frontier

Figure 2: Estimates and standard errors along the L1 frontier with the GOTV data. Thevertical red dashed line is the point at which all controls in strata without common em-pirical support have been dropped. The horizontal blue line is the point estimate from theinstrumental variables estimator.

imbalance metrics.)

Next, in Figure 4, we plot the effect sizes along the Mahalanobis frontier. As in

the L1 example, we recover the experimental result after dropping the most imbalanced

observations.

5.2 Lalonde Data

We now analyze data from the National Supported Work Demonstration and the Current

Populaton Survey (LaLonde, 1986; Dehejia and Wahba, 2002). The National Supported

Work Demonstration was an experimental intervention, while the Current Population Sur-

vey is observational data sometimes appended to the experiment and treated as “controls”

as a means of testing a method’s ability to recover the experimental effect. We use the

data in this way, and, as in the GOTV example, display the experimental estimate as a

blue line on the effect plot.

As is standard in the use of these data, we match on age, education, race (black or

Hispanic), marital status, whether or not the subject has a college degree, earnings in

18

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

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●●

●● ● ● ● ● ● ● ● ●0.0

0.5

1.0

1.5

0 1000 2000 3000Number of Control Units Pruned

Mah

alan

obis

Val

ue

Mahalanobis Imbalance Frontier with GOTV Data

Figure 3: The L1 frontier with the GOTV data.

1974, and earnings in 1975. Earnings in 1978 is the outcome variable. In contrast to

the GOTV experiment, for the Lalonde frontier, we drop both treated and control units

and thus let the quantity of interest change (and summarize the changes below). The L1

frontier is displayed in Figure 5.

Next, in Figure 6 we plot the causal effects along the frontier. It is clear that the

difference between the nonexperimental and the experimental results is due again to strata

without common support. Once these observations have been dropped, our estimates are

very similar to those from the sample restricted to experimental units.

In this example, we drop both control and treated units, thus changing the quantity

of interest along the frontier. A crucial step, therefore, is to determine what quantity is

being estimated. To visualize this change, in Figure 7, we plot the means of each variable

(standardized to the unit interval) along the entire frontier. Relatively small changes in

the quantity of interest based on this measure occur until more than 5000 observations are

pruned. After about 12,000 the changes are rapid and large.

19

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●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●

●●●●●●●●●●●●●●

●

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●

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● ●

●

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●

●

−0.05

0.00

0.05

0 1000 2000 3000Number of Control Units Pruned

Effe

ct S

ize

and

SE

s

Effects with GOTV Data Along Mahal Frontier

Figure 4: Estimates and standard errors along the Mahalanobis frontier with the GOTVdata. The blue horizontal line is the point estimate from the instrumental variables esti-mator.

5.3 Nielsen et al

In our final example, we examine data estimating the effect of foreign aid shocks on

civil conflict propensity (Nielsen et al., 2011). Unlike the previous two illustrations, in

this case there exists no experimental comparison. However, the reported results used

propensity score matching to account for the potential selection problem — donors are

likely to decrease aid if they observe signs of an impending civil war — and find that aid

shocks increase the propensity for civil war within both the matched and the full sample.

However, as is customary, the original analysis generated a single matched data set and

made inferences on its basis.

By making inferences all along the full frontier, we provide a great deal of additional

information. Moreover, the matched data set used in the published paper was not the

best-balanced data set of its size, by either metric. Our ability to return the best-balanced

sample conditional on size is the right thing to do methodologically, and adds to the anal-

ysis of this data. Were we only able to return a single point on the frontier, ours would

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0.25

0.50

0.75

1.00

0 5000 10000 15000Number of Observations Pruned

L1 V

alue

L1 Imbalance Frontier with Lalonde Data

Figure 5: The L1 frontier with the Lalonde data.

still represent an improvement on existing techniques. And by making estimates along

the full range of the frontier, we do much more than that.

In this particular example, we calculate the Mahalanobis frontier and find that the

substantive conclusion (foreign aid shocks increase civil war propensity) remains strong

across the full range of the frontier. A finding of this sort is evidence in favor of the paper’s

claim, for it demonstrates a robustness not evident in the original analysis.

The authors used rare events logistic regression (King, 2001), and we follow suit. We

use their model specification and match on the covariates they used to generate propen-

sity scores. The Mahalanobis imbalance frontier is displayed in Figure 8 and the effects

are displayed in Figure 9. We only display the range over which rare events logit is rea-

sonable. In the published results with the matched sample, the estimated effect of an aid

shock on civil war propensity is 0.934 (log-odds), with a standard error of 0.318.

In this example, we drop both treated and control units, thus allowing the quantity of

interest to vary as we drop observations. The means of each covariate are displayed in

Figure 10.

Although we use FSATT for this illustration, allowing the matching algorithm to de-

21

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●●●

1000

1500

2000

0 5000 10000 15000Number of Observations Pruned

Effe

ct S

ize

and

SE

s

Treatment Effects with Lalonde Data Along L1 Frontier

Figure 6: The L1 frontier with the Lalonde data (both experimental and nonexperimental).The red dotted line is the point at which all observations (treated and control) in stratawithout common empirical support have been dropped.

fine the quantity of interest, the means of the covariates remain largely stable across the

entire frontier. Although higher moments could be considered, this suggests that the quan-

tity of interest is remaining roughly the same across the entire frontier. Even as treated and

control units that are least balanced are removed, the overall distribution of the covariates

in the matched samples remains about the same. This means that although the results are

likely to generalize fairly well to the overall sample of potential civil war onset cases,

even as the matched samples we analyze become more and more balanced. Of course,

this need not always be the case and care must be taken to specify the specific types of

units for which the findings apply.

Substantively, the original propensity score analysis estimated a logit coefficient of

approximately 0.9; the estimates we obtain along the frontier range from 0.9 to 0.45. All

are positive, supporting the original conclusions that aid shocks cause an increased risk of

violent conflict. However, not all estimates are statistically significant. The loss of signif-

icance is due to the decreasing magnitude of the effect size, not the widening of standard

22

0.2

0.4

0.6

0 5000 10000 15000Number of Observations Pruned

Sta

ndar

dize

d M

ean

Val

ue

VariableAgeBlackEducationHispanicIncome in '74Income in '75MarriedNo degree

Change in Means Across the Lalonde L1 Frontier

Figure 7: The means of each covariate along the Lalonde L1 frontier, standardized torange from zero to one.

errors from decreased sample size. This means that the magnitude of the causal effect

may be smaller than previously reported. We also note that there are substantial jumps in

the estimated effect size along the frontier that are not mirrored by similar changes in the

means of the covariates. Of course, analysts who have different preferences for trading

off balance and sample size may arrive at different conclusions.

6 Concluding RemarksMatching methods are at once very simple and easy to understand and also provide a

powerful nonparametric way to control for measured covariates. However, in practice,

implementing matching methods had become arduous and often frustrating because they

optimize with respect to one imbalance measure or the matched sample size and then

are evaluated with respect to another measure or sample size. The matching frontier we

describe here offers simultaneous optimization of both criteria while retaining as much of

the simplicity that made matching attractive in the first place.

23

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3

6

9

0 1000 2000Number of Observations Pruned

L1 V

alue

Mahal Imbalance Frontier with Nielsen Data

Figure 8: The Mahalanobis frontier with data from Nielsen et al. (2011)

ReferencesArceneaux, Kevin, Alan S Gerber and Donald P Green. 2006. “Comparing experimental

and matching methods using a large-scale voter mobilization experiment.” PoliticalAnalysis 14(1):37–62.

Austin, Peter C. 2008. “A critical appraisal of propensity-score matching in the medicalliterature between 1996 and 2003.” jasa 72:2037–2049.

Austin, Peter C. 2009. “Some Methods of Propensity-Score Matching had Superior Per-formance to Others: Results of an Empirical Investigation and Monte Carlo simula-tions.” Biometrical Journal 51(1, February):171–184.

Caliendo, Marco and Sabine Kopeinig. 2008. “Some Practical Guidance for the Imple-mentation of Propensity Score Matching.” Journal of Economic Surveys 22(1):31–72.

Crump, Richard K., V. Joseph Hotz, Guido W. Imbens and Oscar Mitnik. 2009. “Dealingwith limited overlap in estimation of average treatment effects.” Biometrika 96(1):187.

Dehejia, Rajeev H and Sadek Wahba. 2002. “Propensity score-matching methods fornonexperimental causal studies.” Review of Economics and statistics 84(1):151–161.

Diamond, Alexis and Jasjeet S Sekhon. 2012. “Genetic matching for estimating causaleffects: a general multivariate matching method for achieving balance in observational

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0 500 1000 1500Number of Observations Pruned

Effe

ct S

ize

and

SE

s

Effects with Nielsen Data Along Mahal Frontier

Figure 9: Estimates and standard errors from rare events logit along the Mahalanobisfrontier from Figure 8.

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